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Puerto Rico Team Selection Test
2022 Puerto Rico Team Selection Test
6
|f(x)-f(y)|<=2022 Puerto Rico TST 2022.1.6
|f(x)-f(y)|<=2022 Puerto Rico TST 2022.1.6
Source:
March 23, 2024
algebra
inequalities
Problem Statement
Let
f
f
f
be a function defined on
[
0
,
2022
]
[0, 2022]
[
0
,
2022
]
, such that
f
(
0
)
=
f
(
2022
)
=
2022
f(0) = f(2022) = 2022
f
(
0
)
=
f
(
2022
)
=
2022
, and
∣
f
(
x
)
−
f
(
y
)
∣
≤
2
∣
x
−
y
∣
,
|f(x) - f(y)| \le 2|x -y|,
∣
f
(
x
)
−
f
(
y
)
∣
≤
2∣
x
−
y
∣
,
for all
x
,
y
x, y
x
,
y
in
[
0
,
2022
]
[0, 2022]
[
0
,
2022
]
. Prove that for each
x
,
y
x, y
x
,
y
in
[
0
,
2022
]
[0, 2022]
[
0
,
2022
]
, the distance between
f
(
x
)
f(x)
f
(
x
)
and
f
(
y
)
f(y)
f
(
y
)
does not exceed
2022
2022
2022
.
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